We are given the following joint distribution:

X123411/4 0 1/8 1/8 Y21/8 0 1/16 1/16 31/16 0 1/32 1/32 41/16 0 1/32 1/32

(The X and Y are labels and should span the entire row/column, but I can't figure out how to merge cells with this).

Out of the questions asked regarding this table, I was stuck on three.

1) What isP(X)?

I interpreted this as "what is the probability that X occurs", but isn't that the sum of all of these probabilities? And therefore, 1? It seems incorrect.

2) What it P(Y|X)?I'm getting confused about what is being asked when they simply ask things about X and Y without mentioning a specific condition for one of them. For example, if they asked for P(X|Y=3), I know that as the marginal probability and would sum up P(1,3)+P(2,3)+P(3,3)+P(4,3). But what does it mean when it's just something like P(Y|X)? I know that in English it's ready as "probability of Y given X", but without some sort of condition it doesn't make sense to me.

3) What is P(X=3|Y)?I'm confused about this because the conditional (=3) is in the first argument. As noted above, if it's the other way around I know how to solve it. But I'm not sure what it is asking here. Is it simply asking for P(3,1)+P(3,2)+P(3,3)+P(3,4) like a regular marginal? I'm confused about what it means to be "given Y" in such a circumstance.

Thanks.